Let $n$ be a positive integer and $m$ be a positive even integer. Let${\mathcal A}$ be an $m^{th}$ order $n$-dimensional real weakly symmetrictensor and ${\mathcal B}$ be a real weakly symmetric positive definite tensorof the same size. $\lambda \in R$ is called a ${\mathcal B}_r$-eigenvalue of${\mathcal A}$ if ${\mathcal A} x^{m-1} = \lambda {\mathcal B} x^{m-1}$ forsome $x \in R^n \backslash \{0\}$. In this paper, we introduce twounconstrained optimization problems and obtain some variationalcharacterizations for the minimum and maximum ${\mathcal B}_r$--eigenvalues of${\mathcal A}$. Our results extend Auchmuty's unconstrained variationalprinciples for eigenvalues of real symmetric matrices. This unconstrainedoptimization approach can be used to find a Z-, H-, or D-eigenvalue of an evenorder weakly symmetric tensor. We provide some numerical results to illustratethe effectiveness of this approach for finding a Z-eigenvalue and fordetermining the positive semidefiniteness of an even order symmetric tensor.
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机译：假设$ n $为正整数，$ m $为正偶数。设$ {\ mathcal A} $是一个$ m ^ {th} $阶$ n $维实弱对称张量，$ {\ math B} $是一个相同大小的实弱对称正定张量。如果$ {\ mathcal A} x ^ {m-1} = \ lambda {\ mathcal B}，则$ \ lambda \ in R $被称为$ {\ mathcal B} _r $-本征值$ {\ mathcal A} $ x ^ {m-1} $大约$ x \ in R ^ n \反斜杠\ {0 \} $。在本文中，我们介绍了两个无约束的优化问题，并获得了最小和最大$ {\ mathcal B} _r $-$ {\ mathcal A} $的特征值的一些变分特征。我们的结果扩展了Auchmuty的实对称矩阵特征值的无约束变分原理。这种不受约束的优化方法可用于找到偶数次弱对称张量的Z，H或D特征值。我们提供了一些数值结果，以说明该方法对于寻找Z特征值并确定偶数阶对称张量的正半定性的有效性。
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